I recently meet the following two weird "Fourier transform" questions.
(I), Suppose that $F$ is a $p$-adic field (the same question can be asked over any local field, including $\mathbb{R}$ and $\mathbb{C}$) and $\psi$ be a fixed nontrivial additive character of $F$. Let $W$ be a function on $GL_2(F)$ which satisfies the following 2 conditions:
(1), $W\left(\begin{pmatrix} 1& x \\ &1 \end{pmatrix} g\right)=\psi(x)W(g)$, for all $x\in F, g\in GL_2(F)$, and
(2), there exists an open subgroup $K$ of $GL_2(F)$ such that $W(gk)=W(g)$ for all $g\in GL_2(F), k\in K$.
One can think $W$ as a Whittaker function of a smooth representation of $GL_2(F)$. One extend the definition of $W$ to all $2\times 2$ matrices by zero extension, i.e., if $g\in \mathrm{Mat}_{2\times 2}(F)$ and $\det(g)=0$, then define $W(g)=0$.
Let $\mathcal{S}(F^2)$ be the space of Bruhat-Schwartz functions on $F^2$. Fix a nonzero Whittaker function $W$ as above. For $\phi\in \mathcal{S}(F^2)$, define $$\widehat{\phi}(x_1,x_2)=\int_{F^2}W\left( I_2+\begin{pmatrix} x_1\\ x_2 \end{pmatrix}\begin{pmatrix} y_1 & y_2\end{pmatrix}\right)\phi(y_1,y_2)dy_1 dy_2.$$ Here $I_2$ is the $2\times 2$ identity matrix and $W$ is omitted from the notation $\widehat{\phi}$.
Question (I): Do we know that $\widehat \phi\in \mathcal{S}(F^2)$ and $\phi\mapsto \widehat\phi$ defines an isomorphism?
This looks like a Fourier transform. Also, it is easy to see that the function $x_1\mapsto \widehat{\phi}(x_1,0)$ has compact support. But I don't know how to prove the above statement in general.
(II) The second question involves a Fourier transform in a similar flavor, but not the same with the above one. Suppose that $k$ is a finite field and $\psi$ is a nontrivial additive character of $k$. Let $\mathscr{B}$ be the set of functions $B$ on $GL_2(k)$ such that $$B\left(\begin{pmatrix}1&x\\ &1 \end{pmatrix} g \begin{pmatrix}1&y\\ &1 \end{pmatrix}\right)=\psi(x)\psi(-y)B(g), \forall x,y\in k, g\in GL_2(k).$$ One can think such a function $B$ as a Bessel function. For $B\in \mathscr{B}$, define \begin{align*}&\mathcal{F}_B\left(\begin{pmatrix} x_{11} & x_{12}\\ x_{21}& x_{22} \end{pmatrix}\right)\\ &\qquad :=\sum_{s_1,s_2,r_1,r_2\in k} \psi(x_{22}r_1+x_{21}r_2+s_1)B\left(\begin{pmatrix} x_{11} & x_{12}\\ x_{21}& x_{22} \end{pmatrix} \left(I_2+\begin{pmatrix} s_2\\ -s_1 \end{pmatrix}\begin{pmatrix} -r_1 & r_2\end{pmatrix} \right) \right), \end{align*} where if $\det(g)=0$, we view $B(g)=0.$
It is not hard to show that $\mathcal{F}_B\in \mathscr{B}$. The negative signs in the definition of $\mathcal{F}_B$ is to make sure $\mathcal{F}_B\in \mathscr{B}.$
Question (II): Is it true that $B\mapsto \mathcal{F}_B$ is a bijection?
If the answer of (II) is affirmative, I am also wondering if there is a local field analogue.
Question (III): Are there generalizations to $GL_n$?
Thanks in advance.
